## Abstract

The second author proved in [7] that each cartesian closed category

of pointed domains and Scott-continuous functions is contained in either

the category of Lawson-compact domains or that of L-domains, and this

result eventually led to a classification of continuous domains with respect

to cartesian closedness, as laid out in [8].

In this paper, we generalise this result to the category LcS of pointed

locally compact sober dcpos and Scott-continuous functions, and show

that any cartesian closed full subcategory of LcS is contained in either

the category of stably compact dcpos or that of L-dcpos. (Note that

for domains Lawson-compactness and stable compactness are equivalent.)

As we will show, this entails that any candidate for solving the Jung-Tix

problem in LcS must be stably compact.

To prove our dichotomy result, we first show that any dcpo with a core-

compact function space must be meet-continuous; then we prove that a

function space in LcS is meet-continuous only if either its input dcpo is

coherent or its output dcpo has complete principal ideals.

of pointed domains and Scott-continuous functions is contained in either

the category of Lawson-compact domains or that of L-domains, and this

result eventually led to a classification of continuous domains with respect

to cartesian closedness, as laid out in [8].

In this paper, we generalise this result to the category LcS of pointed

locally compact sober dcpos and Scott-continuous functions, and show

that any cartesian closed full subcategory of LcS is contained in either

the category of stably compact dcpos or that of L-dcpos. (Note that

for domains Lawson-compactness and stable compactness are equivalent.)

As we will show, this entails that any candidate for solving the Jung-Tix

problem in LcS must be stably compact.

To prove our dichotomy result, we first show that any dcpo with a core-

compact function space must be meet-continuous; then we prove that a

function space in LcS is meet-continuous only if either its input dcpo is

coherent or its output dcpo has complete principal ideals.

Original language | English |
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Pages (from-to) | 935-951 |

Journal | Houston Journal of Mathematics |

Volume | 45 |

Issue number | 3 |

Publication status | Published - 30 Nov 2019 |